To simplify the expression \(\frac{2^{2}x^{-1}y^{-8}}{9^{-1}x^{0}y^{-8}}\), let's analyze it step by step.
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Break down the terms:
- The numerator is \(2^{2} \cdot x^{-1} \cdot y^{-8}\).
- The denominator is \(9^{-1} \cdot x^{0} \cdot y^{-8}\).
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Simplify \(x^{0}\):
- Recall that any number raised to the exponent \(0\) is \(1\), so \(x^{0} = 1\).
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Rewrite the expression: The expression now becomes: \[ \frac{2^{2} \cdot x^{-1} \cdot y^{-8}}{9^{-1} \cdot 1 \cdot y^{-8}} = \frac{2^{2} \cdot x^{-1} \cdot y^{-8}}{9^{-1} \cdot y^{-8}} \]
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Cancel \(y^{-8}\):
- Since \(y^{-8}\) in the numerator and denominator can be canceled out (assuming \(y \neq 0\)), we are left with: \[ \frac{2^{2} \cdot x^{-1}}{9^{-1}} = 2^{2} \cdot x^{-1} \cdot 9 \]
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Simplify the expression:
- \(9^{-1} = \frac{1}{9}\), so when we multiply it back in: \[ 2^{2} \cdot x^{-1} \cdot 9 = 4 \cdot 9 \cdot x^{-1} = \frac{36}{x} \]
The simplified expression is: \[ \frac{36}{x} \]